The song was entered in the Register of the Stationers' Company in 1588, licensed to Thomas Orwyn. There is no particular reason to think it any older than that. The alleged reference to Henry VIII is not mentioned by Chappell and is likely a modern myth.

Ravenscroft's text has "Clouting of". The original meaning would almost certainly be either patching or, most likely, repairing or reinforcing the soles with nails (clout-nails, in fact).

Earlier discussions and DT examples can be found via the search engine. Note, though, that the information given with the DT file WHO'S THE FOOL NOW or MARTIN SAID TO HIS MAN is misleading; the text is not the one printed by Chappell, but a modern collation of verses from various sources including Ravenscroft, some altered to fit.